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Properties

Label 2-161-161.160-c3-0-21
Degree $2$
Conductor $161$
Sign $0.991 + 0.129i$
Analytic cond. $9.49930$
Root an. cond. $3.08209$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.765·2-s + 0.826i·3-s − 7.41·4-s − 5.17·5-s + 0.633i·6-s + (15.9 − 9.38i)7-s − 11.8·8-s + 26.3·9-s − 3.96·10-s + 51.1i·11-s − 6.12i·12-s − 79.0i·13-s + (12.2 − 7.18i)14-s − 4.28i·15-s + 50.2·16-s + 111.·17-s + ⋯
L(s)  = 1  + 0.270·2-s + 0.159i·3-s − 0.926·4-s − 0.463·5-s + 0.0430i·6-s + (0.862 − 0.506i)7-s − 0.521·8-s + 0.974·9-s − 0.125·10-s + 1.40i·11-s − 0.147i·12-s − 1.68i·13-s + (0.233 − 0.137i)14-s − 0.0736i·15-s + 0.785·16-s + 1.58·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.129i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.991 + 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.991 + 0.129i$
Analytic conductor: \(9.49930\)
Root analytic conductor: \(3.08209\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 161,\ (\ :3/2),\ 0.991 + 0.129i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.69023 - 0.109814i\)
\(L(\frac12)\) \(\approx\) \(1.69023 - 0.109814i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-15.9 + 9.38i)T \)
23 \( 1 + (-87.0 - 67.7i)T \)
good2 \( 1 - 0.765T + 8T^{2} \)
3 \( 1 - 0.826iT - 27T^{2} \)
5 \( 1 + 5.17T + 125T^{2} \)
11 \( 1 - 51.1iT - 1.33e3T^{2} \)
13 \( 1 + 79.0iT - 2.19e3T^{2} \)
17 \( 1 - 111.T + 4.91e3T^{2} \)
19 \( 1 - 73.3T + 6.85e3T^{2} \)
29 \( 1 - 25.0T + 2.43e4T^{2} \)
31 \( 1 + 246. iT - 2.97e4T^{2} \)
37 \( 1 + 207. iT - 5.06e4T^{2} \)
41 \( 1 - 185. iT - 6.89e4T^{2} \)
43 \( 1 - 455. iT - 7.95e4T^{2} \)
47 \( 1 - 298. iT - 1.03e5T^{2} \)
53 \( 1 + 614. iT - 1.48e5T^{2} \)
59 \( 1 + 583. iT - 2.05e5T^{2} \)
61 \( 1 - 63.2T + 2.26e5T^{2} \)
67 \( 1 - 258. iT - 3.00e5T^{2} \)
71 \( 1 + 548.T + 3.57e5T^{2} \)
73 \( 1 - 864. iT - 3.89e5T^{2} \)
79 \( 1 + 499. iT - 4.93e5T^{2} \)
83 \( 1 + 513.T + 5.71e5T^{2} \)
89 \( 1 - 580.T + 7.04e5T^{2} \)
97 \( 1 - 206.T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.65147615461397412528155490641, −11.51311049742474512158368944641, −10.06670884801565476933069578209, −9.689331288693066489414193283730, −7.79678799250588691855671979818, −7.65198663314995625088736714307, −5.45423354891565423337384081291, −4.60023170606183794007552833858, −3.51936070870321416411205667330, −1.05446536838679007736306464843, 1.19442235620439102714448981641, 3.47332585901558098882237999977, 4.63119556598245536997591437332, 5.69933781088549968220120496674, 7.26755183739099746942424061906, 8.416902010027509854750027305558, 9.160431007278312542619880726123, 10.43718648604952277115849082269, 11.82611710217535255804835830504, 12.20464280360728296978642386647

Graph of the $Z$-function along the critical line